Updated 7:02 p.m. | Senate Minority Whip John Cornyn told reporters Monday that it would be "premature" to move on a standalone guarantee of back pay for federal workers without addressing other elements of the government shutdown.

"It think it's really premature to be dealing with that until we deal with the underlying problem," the Texas Republican said. "We've offered a number of bills to try to alleviate some of the hardship, and ... they've been swatted down out of hand."

The GOP-led House has passed a number of narrowly-tailored stopgap spending bills that would restore funding for an assortment of federal government programs, including the National Institutes of Health and the Federal Emergency Management Agency.

President Barack Obama visited FEMA on Monday.

"Thanks to the folks at FEMA, we were prepared for what might have happened down in Florida. Nevertheless, the government is still shut down, services are still interrupted and hundreds of thousands of hardworking public servants, including many FEMA professionals, are still furloughed without pay, or they’re not allowed to work at all," Obama said.

Cornyn indicated that Republicans should be allowed to offer amendments to the measure regarding retroactive pay for furloughed federal employees, which the House passed Saturday, 407-0, as part of the larger strategy. Such a move could give GOP senators a chance to try to force votes on expanding the scope of the measure.

"I think if Sen. [Harry] Reid wants to bring that to the floor, he should bring it to the floor, and we should, it should be subject to the normal legislative process," Cornyn said. "We ought to legislate."

Cornyn said that he did not expect the majority leader to call the measure up, however.

Advised of the procedural situation, Maine Republican Sen. Susan Collins expressed disappointment that the federal worker back pay measure might not be expedited.

"I think that's unfortunate. I think it would be very reassuring to constituents across the country who are out of work through no fault of their own if they at least knew that they were going to receive retroactive pay, since what this represents is not a failure on their part, but a failure on the part of Congress and the White House," Collins said.

Meanwhile, Senate Minority Leader Mitch McConnell said in his opening remarks on the Senate floor Monday that the chamber should move on the small continuing resolutions, a path Democrats have rejected, saying it would amount to cherry-picking which agencies get funded.

"There's actually a fair amount of agreement among Republicans and Democrats over there that lawmakers have a duty and responsibility that rises above the politics of the moment to fund things like veterans, cancer trials, the National Guard and Reservists in every state," the Kentucky Republican said.